Some dis-analogies between Two Basic Quantities (Space and Time) in Newton's Principia

Thus far it has seemed best to explain the senses in which less familiar words are to be taken in this treatise. Although time, space, place, and motion are very familiar to everyone, it must be noted that these quantities are popularly conceived solely with reference to the objects of sense perception. And this is the source of certain preconceptions; to eliminate them it is useful to distinguish these quantities into absolute and relative, true and apparent, mathematical and common.

1. Absolute, true, and mathematical time, in and of itself and of its own nature, without reference to anything external, flows uniformly and by another name is called duration. Relative, apparent, and common time is any sensible and external measure (precise or imprecise) of duration by means of motion; such a measure—for example, an hour, a day, a month, a year—is commonly used instead of true time.

2. Absolute space, of its own nature without reference to anything external, always remains homogeneous and immovable. Relative space is any movable measure or dimension of this absolute space; such a measure or dimension is determined by our senses from the situation of the space with respect to bodies and is popularly used for immovable space, as in the case of space under the earth or in the air or in the heavens, where the dimension is determined from the situation of the space with respect to the earth. Absolute and relative space are the same in species and in magnitude, but they do not always remain the same numerically. For example, if the earth moves, the space of our air, which in a relative sense and with respect to the earth always remains the same, will now be one part of the absolute space into which the air passes, now another part of it, and thus will be changing continually in an absolute sense.--Newton, Scholium to the Definitions, Principia, translated by Whitman & Cohen) 1999: 408-409).

My present interest is to articulate four dis-analogies in the paragraphs labeled, '1' and '2'.* For the purposes of analysis, I distinguish in Newton's thought between four basic quantities and three fundamental distinctions that Newton applies to these four basic quantities. The four basic quantities are “time, space, place, and motion." The three fundamental distinctions are "absolute and relative';' "true and apparent;" and "mathematical and common.” In what follows I focus only on basic quantities. Here I stipulate that such a quantity is abstract (see Smeenk & Schliesser 2017). And my interest here is primarily in the significance of the measures of two such basic quantities (time and space).

The first dis-analogy is related to the first sentence of each paragraph: ‘true, and mathematical’ are present as modifications of (absolute) time and absent as modifications of (absolute) space. The second is related to the second sentence of each paragraph: that “apparent, and common” are present as modification of (relative) time and absent as modification of (relative) space. Katherine Brading (2017) and I (see here in 2013) have given different interpretations of the significance of these two dis-analogies, but I will bracket that today.

Third, there is an asymmetry in the measures Newton proposed. In order to measure time (an abstract quantity) Newton proposes to use the motion of bodies. However, in order to measure space (another abstract quantity) Newton proposes to use the “situation” of a part of space to another part. Crucially, the measure of space is identical in “species and in magnitude” to the thing it is a measure of (space). This identity (in species and magnitude) is omitted in the measure of time. Motions of bodies, even the regular motions of the solar system or pendulum clocks,  are not identical in species and magnitude to duration.

Brading (2019) explains what’s going on in the third dis-analogy. In her terminology, rods (that is, measures of space) are geometrical just as space is. Whereas clocks (that is, a regular motion of bodies) are, for Newton, dynamical systems. And whatever time “in and of itself and of its own nature,” might be it is not natural to think of it as a dynamical system. That is to say, there seems to be no possible gap between the measure(s) of space and space (Brading 2019: 160-161); there does seem a possible gap between the measure(s) of time and time (Brading 2019: 162).  

The reason why clocks are a dynamical system on Newton’s account is very nicely explained by Brading:

“It is central to the project of the Principia,” she writes, “that forces and the motions of bodies are inter-dependent. Newton emphasizes this in the Preface to the first edition:

For the basic problem of philosophy seems to be to discover the forces of nature from the phenomena of motions and then to demonstrate the other phenomena from these forces (Newton, 1999, p. 382)

This means that all clocks in the Principia are dynamical systems, because they are systems of bodies in motion, and motions and forces are inter-related,” (Brading 2019: 162).”

That there is a possible gap between the measure of time and time in of itself is not just caused by the fact that the measure is a dynamical system and it seems odd to say this about time itself (where forces and bodies are absent), but also, and more important, because while time “flows uniformly” according to Newton’s stipulation (Brading 2019: 164), there is no guarantee that any measure, which is a natural process of bodies in motion, does so. And even when one constructs an artificial measure (a pendulum clock) or an abstract measure (a mathematical equation of time), there is no guarantee that its regularity and time’s uniformity are identical (Brading 2017: 34).

Now, it worth noting that Brading's use of "rods" is anachronistic because Newton uses "situations." Unfortunately, Newton does not explain what a 'situation' is. I think we can infer it is a kind of mental inspection or abstraction from other phenomenal details. Obviously, if you want to stabilize situations, rods are tempting instruments. 

As an aside, in his Essay, Locke, who I am treating here as an independent guide on contemporary use (not a source), treats 'situation' as an ingredient of complex ideas of (solid) bodies (2.23.9) and sometimes as a way to convey a relationship among sensible parts (2.4.4). 

The more important point, and here I am drawing on a note from Chris Smeenk is that "situation of the space with respect to bodies” (August 28, 2020) involves bodies apart in some sense; because unoccupied spaces are invisible and so difficult to use as markers of situations. Now, Smeenk worries, I think, this reintroduces time or dynamics. If that is so, then the gap that Brading diagnoses on the measure of time vs time side, may also exist on the space vs measure of space side. 

The reason why I think that there need not be such a gap between the measure of space and space is that one can evaluate/inspect a situation at an instance if a situation is small enough. I say this for two reasons: (a) in the early modern period it is often thought that some ideas are secure, or adequate, if they can be inspected at once  or instantaneously (without intermediate relata).** And perhaps Newton thinks that some relatively small situations can function as foundational measures in this way such that situations and spaces are the same in "species and in magnitude." And (b) keeping situations small echoes the manner in which pendulums (a body in motion) can be a reliable measure of time (by keeping the arc small, as was well known to Newton and Huygens)

This (b) also helps qualify the nature of the gap between the measure of time and time diagnosed by Brading. (What follows is also indebted to Smeenk.) For, once the ideal time keeper has been mathematically articulated and shown to be possible, Newton can bound the amount of error associated with departures from this ideal case. In fact, from the mid 1680s Huygens shows how to start doing this for a pendum in practice. And so that the right thing to say here, and this is very much in the spirit of George Smith's teaching on Newton (which Brading, Smeenk, and I share as common ground), is that Newton also creates a research program of successive approximation into theoretically and empirically establishing the error bounds and methods for correcting them among our measures; in Smeenk's terms of "systematically improving time measurements to approach the ideal of a truly periodic system." (And so Newton helps initiate a research project into measure-theory.)+ And so, if there is a gap, there is also a forward-looking attempt to learn how to recognize and close it over (ahh) time.

The fourth dis-analogy is that the measure of time (bodies in motion) presupposes space or its measure while the measure of space (a situation) does not presuppose time (if the situation can be accessed at an instant). This point is clearly something Newton had reflected on. For in his earlier criticism of Descartes’ physics, Newton argues in De Gravitatione that in order to be able to conceptualize and analyze motion, one must make reference to some “motionless being such as extension alone or space in so far as it is seen to be truly distinct from bodies,” (Newton 2004: 35) One need not agree with Newton’s metaphysics here, to see that some fixed or ordered coordinate system, independent of the bodies, is required for the analysis of motion (with a starting place and a subsequent places, etc.).

 

 

*There are many more metaphysical dis-analogies between space and time in Newton's philosophy (see my 2013 handbook article; Gorham 2011; Gorham & Slowik 2014). 

**If they can't be inspected instantaneously, time and hidden commitments to simultaneity enter in through the back-door. This is something Smeenk worries about. Because I take Newton to be developing an idea of a 'temporal frame,' within which temporal relations are shared within the solar system, I am not worried here about Newtonian over-reliance on simultaneity. But it is clear that as situations become unstable or far apart it is by no means obvious that Newton can prevent a gap from opening up (of the sort that worries Smeenk).

+There is an unpublished presentation, "Indirect (i.e., derived) measurement and Evidence" by George Smith in honor of Patrick Suppes' 90th birthday that has influenced me directly.